where \(Q\)  is an orthonormal matrix [1]}, i.e. \(Q^T Q = Q Q^T = I\) . Note that the transformed matrix \(L_2\)  can be any symmetric positive semi-definite matrix which may not be of Laplacian matrix structure. In order to get a distinct Laplacian matrix on applying similarity transformation to a base Laplacian matrix, we need to search for a special class of orthonormal matrices that not only preserves the eigenvalues of a Laplacian matrix but also preserves its properties and structure.
